This Element takes a deep dive into Goedel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Goedel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
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Worked examples or Exercises
Maße
Höhe: 229 mm
Breite: 152 mm
Dicke: 6 mm
Gewicht
ISBN-13
978-1-108-98699-1 (9781108986991)
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Schweitzer Klassifikation
Introduction; 1. The first version of the proof; 2. Goedel's 'intuitionistically acceptable' second proof of the First Incompleteness Theorem; 3. The unprovability of consistency; 4. Loeb conditions and adequacy; 5. Other proofs of the First and Second Theorems; 6. Mathematical Incompleteness; 7. Set Theoretical Incompleteness; 8. Further philosophical consequences of the Incompleteness Theorems.
